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Re: Re: Proving inequalities with Mathematica
*To*: mathgroup at smc.vnet.net
*Subject*: [mg52553] Re: [mg52499] Re: [mg52491] Proving inequalities with Mathematica
*From*: DrBob <drbob at bigfoot.com>
*Date*: Wed, 1 Dec 2004 05:59:21 -0500 (EST)
*References*: <200411290622.BAA27997@smc.vnet.net> <200411301024.FAA01299@smc.vnet.net>
*Reply-to*: drbob at bigfoot.com
*Sender*: owner-wri-mathgroup at wolfram.com
I wish I'd thought of that. Brilliant, Andrzej.
FWIW, here's a process for making other examples (now that you've pointed the way):
Clear[a,b]
a[n_]=(100/99)^-n;
Simplify[Abs[a[n+1]/a[n]]<1,n>1]
True
(If that fails, choose another series.)
Now choose a[1] and a[2]:
s = Simplify[Sum[a[n], {n, 3, Infinity}]];
{a[1], a[2]} = {x, y} /. First@FindInstance[{x + y == -s, Abs@x > Abs@y >
Abs@a@3}, {x, y}, Reals]
{-(490001/5000), 9703/10000}
All premises are guaranteed at this point (setting b == a), but the conclusion can NEVER be true, since
Sum[a[i]^2, {i, 1, Infinity}] < a[1]^2
is equivalent to:
Sum[a[i]^2, {i, 2, Infinity}] < 0
But wait -- could FindInstance fail? Consider this, just for s >= 0:
s =. ;
Reduce[{s >= 0, x + y == -s,
Abs[x] > Abs[y] > Abs[a3],
a3 \[Element] Reals}, {x, y}, Reals]
s > 0 && ((a3 <= -(s/2) &&
x < a3 - s) ||
(Inequality[-(s/2), Less, a3,
LessEqual, 0] &&
(x < a3 - s || -a3 - s < x <
-(s/2))) ||
(0 < a3 < s/2 &&
(x < -a3 - s || a3 - s < x <
-(s/2))) || (a3 >= s/2 &&
x < -a3 - s)) && y == -s - x
The Or in the middle is a run-down of all the ways a3 can stand in relation to +-s/2 and 0; for every case we have a non-empty range in which to choose x and a formula for y. The situation is similar for s<0.
Bobby
On Tue, 30 Nov 2004 05:24:01 -0500 (EST), Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote:
>
> On 29 Nov 2004, at 15:22, Toshiyuki (Toshi) Meshii wrote:
>
>> Hi,
>>
>> I was wondering whether Mathematica is useful for proving a problem of
>> inequality.
>> My problem is as follows:
>>
>> Let An and Bn (n=1,2,3...) be real sequences.
>> Some characteristics of these sequences are known.
>>
>> i) Abs[An+1/An] < 1
>> ii) Abs[Bn+1/Bn] < 1
>> iii) Sum[An, {1,Infinity}] = 0
>> iv) Sum[An*Bn, {1, Infinity}] = alpha (note: a real number)
>>
>> Then I want to prove with Mathematica that
>> 0 < Abs[alpha] < Abs[A1*B1]
>>
>> Does anyone have an idea?
>>
>> -Toshi
>>
>
> First of all, you have written your condition as
>
>> Abs[An+1/An] < 1
>
> Presumably you meant n+1 to be a subscript (if not the condition is
> always false).
> Assuming that, Mathematica can certainly be helpful in proving your
> claim to be false. Here is an example.
>
> We define the sequence A[n] as follows:
>
> A[1] = -1; A[2] = Log[2];
> A[n_] = (-1)^(n - 1)/(n - 1);
>
> You can check that the condition Abs[A[n+1]/A[n]]<1 is always
> satisfied. We also define the sequence B[n] by
> B[i_] := A[i]
> so that the condition
>
> Abs[B[n+1]/B[n]]<1
>
> is satisifed.
>
> We check that (iii) is satisfied:
>
>
> A[1]+A[2]+Sum[A[n],{n,3,Infinity}]//Simplify
>
> 0
>
> Let's now compute alpha:
>
>
> alpha=A[1]*B[1]+A[2]*B[2]+Sum[A[i]*B[i],{i,3,Infinity}]//N
>
>
> 2.12539
>
> Let's see if your inequality holds:
>
>
> Abs[alpha]<Abs[A[1]*B[1]]
>
> False
>
> Oops ...
>
>
> Andrzej Kozlowski
> Chiba, Japan
> http://www.akikoz.net/~andrzej/
> http://www.mimuw.edu.pl/~akoz/
>
>
>
>
--
DrBob at bigfoot.com
www.eclecticdreams.net
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